\newproblem{lay:2_3_33}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 2.3.33}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  Let $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$ given by $T(x_1,x_2)=(-5x_1+9x_2,4x_1-7x_2)$. Show that $T$ is invertible and find a formular for $T^{-1}$.
}{
  % Solution
	We may write the transformation as
	\begin{center}
		$T(x_1,x_2)=\begin{pmatrix}-5 & 9 \\ 4 & -7\end{pmatrix}\begin{pmatrix}x_1\\x_2\end{pmatrix}$
	\end{center}
	By defining the matrix $A=\begin{pmatrix}-5 & 9 \\ 4 & -7\end{pmatrix}$ and computing its inverse
	$A^{-1}=\begin{pmatrix}-0.0986 & 0.1268 \\ 0.0563 & 0.0704\end{pmatrix}$, we may write the inverse transformation as
	\begin{center}
		$T(x_1,x_2)=\begin{pmatrix}-0.0986 & 0.1268 \\ 0.0563 & 0.0704\end{pmatrix}\begin{pmatrix}x_1\\x_2\end{pmatrix}$
	\end{center}
}
\useproblem{lay:2_3_33}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
